Right triangle hypotenuse is basically a geometric mean of hypotenuse sections generated due to perpendicular line from vortex to hypotenuse. But it's developed using Congruency theory among two right triangle generated because of perpendicular on hypotenuse from vortex of hypotenuse.
You can use Pythagorus theorem and little algebra to get your result.
Write the Pythagorus theorem for the original triangle. express the hypotenuse as sum of the two parts. And expand the whole square term. Now write the other two sides using Pythagoras in terms of the perpendicular from the vertex and hypotenuse sections.
Solving these three equations you get the altitude of the triangle as equal to the geometric mean of the two sections of the hypotenuse.
NO congruent triangles or trigonometry used!
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